The Calculus of Retirement Income: Financial Models for Pension Annuities and Life Insurance,9780521842587

The Calculus of Retirement Income: Financial Models for Pension Annuities and Life Insurance

by
ISBN13: 9780521842587
ISBN10: 0521842581
Format: Hardcover
Pub. Date: 2006-03-13
Publisher(s): Cambridge University Press
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Summary

The book introduces and develops the basic actuarial models and underlying pricing of life-contingent pension annuities and life insurance from a unique financial perspective. The ideas and techniques are then applied to the real-world problem of generating sustainable retirement income towards the end of the human life-cycle. The role of lifetime income, longevity insurance, and systematic withdrawal plans are investigated in a parsimonious framework. The underlying technology and terminology of the book are based on continuous-time financial economics by merging analytic laws of mortality with the dynamics of equity markets and interest rates. Nonetheless, the book requires a minimal background in mathematics and emphasizes applications and examples more than proofs and theorems. It can serve as an ideal textbook for an applied course on wealth management and retirement planning in addition to being a reference for quantitatively-inclined financial planners.

Table of Contents

List of Figures and Tables x
I MODELS OF ACTUARIAL FINANCE
1 Introduction and Motivation
3(14)
1.1 The Drunk Gambler Problem
3(2)
1.2 The Demographic Picture
5(4)
1.3 The Ideal Audience
9(1)
1.4 Learning Objectives
10(2)
1.5 Acknowledgments
12(2)
1.6 Appendix: Drunk Gambler Solution
14(3)
2 Modeling the Human Life Cycle
17(17)
2.1 The Next Sixty Years of Your Life
17(1)
2.2 Future Value of Savings
18(2)
2.3 Present Value of Consumption
20(2)
2.4 Exchange Rate between Savings and Consumption
22(4)
2.5 A Neutral Replacement Rate
26(1)
2.6 Discounted Value of a Life-Cycle Plan
27(1)
2.7 Real vs. Nominal Planning with Inflation
28(2)
2.8 Changing Investment Rates over Time
30(2)
2.9 Further Reading
32(1)
2.10 Problems
33(1)
3 Models of Human Mortality
34(30)
3.1 Mortality Tables and Rates
34(1)
3.2 Conditional Probability of Survival
35(2)
3.3 Remaining Lifetime Random Variable
37(1)
3.4 Instantaneous Force of Mortality
38(1)
3.5 The ODE Relationship
39(2)
3.6 Moments in Your Life
41(3)
3.7 Median vs. Expected Remaining Lifetime
44(1)
3.8 Exponential Law of Mortality
45(1)
3.9 Gompertz-Makeham Law of Mortality
46(3)
3.10 Fitting Discrete Tables to Continuous Laws
49(2)
3.11 General Hazard Rates
51(2)
3.12 Modeling Joint Lifetimes
53(2)
3.13 Period vs. Cohort Tables
55(4)
3.14 Further Reading
59(1)
3.15 Notation
60(1)
3.16 Problems
60(1)
3.17 Technical Note: Incomplete Gamma Function in Excel
61(1)
3.18 Appendix: Normal Distribution and Calculus Refresher
62(2)
4 Valuation Models of Deterministic Interest
64(19)
4.1 Continuously Compounded Interest Rates?
64(2)
4.2 Discount Factors
66(1)
4.3 How Accurate Is the Rule of 72?
67(1)
4.4 Zero Bonds and Coupon Bonds
68(2)
4.5 Arbitrage: Linking Value and Market Price
70(2)
4.6 Term Structure of Interest Rates
72(1)
4.7 Bonds: Nonflat Term Structure
73(1)
4.8 Bonds: Nonconstant Coupons
74(1)
4.9 Taylor's Approximation
75(1)
4.10 Explicit Values for Duration and Convexity
76(2)
4.11 Numerical Examples of Duration and Convexity
78(2)
4.12 Another Look at Duration and Convexity
80(1)
4.13 Further Reading
81(1)
4.14 Notation
82(1)
4.15 Problems
82(1)
5 Models of Risky Financial Investments
83(27)
5.1 Recent Stock Market History
83(3)
5.2 Arithmetic Average Return versus Geometric Average Return
86(2)
5.3 A Long-Term Model for Risk
88(3)
5.4 Introducing Brownian Motion
91(6)
5.5 Index Averages and Index Medians
97(1)
5.6 The Probability of Regret
98(2)
5.7 Focusing on the Rate of Change
100(1)
5.8 How to Simulate a Diffusion Process
101(1)
5.9 Asset Allocation and Portfolio Construction
102(2)
5.10 Space Time Diversification
104(3)
5.11 Further Reading
107(1)
5.12 Notation
108(1)
5.13 Problems
108(2)
6 Models of Pension Life Annuities
110(28)
6.1 Motivation and Agenda
110(1)
6.2 Market Prices of Pension Annuities
110(4)
6.3 Valuation of Pension Annuities: General
114(1)
6.4 Valuation of Pension Annuities: Exponential
115(1)
6.5 The Wrong Way to Value Pension Annuities
115(1)
6.6 Valuation of Pension Annuities: Gompertz-Makeham
116(3)
6.7 How Is the Annuity's Income Taxed?
119(2)
6.8 Deferred Annuities: Variation on a Theme
121(2)
6.9 Period Certain versus Term Certain
123(2)
6.10 Valuation of Joint and Survivor Pension Annuities
125(3)
6.11 Duration of a Pension Annuity
128(2)
6.12 Variable vs. Fixed Pension Annuities
130(4)
6.13 Further Reading
134(2)
6.14 Notation
136(1)
6.15 Problems
136(2)
7 Models of Life Insurance
138(26)
7.1 A Free (Last) Supper?
138(1)
7.2 Market Prices of Life Insurance
138(1)
7.3 The Impact of Health Status
139(1)
7.4 How Much Life Insurance Do You Need?
140(2)
7.5 Other Kinds of Life Insurance
142(1)
7.6 Value of Life Insurance: Net Single Premium
143(2)
7.7 Valuing Life Insurance Using Pension Annuities
145(2)
7.8 Arbitrage Relationship
147(1)
7.9 Tax Arbitrage Relationship
148(1)
7.10 Value of Life Insurance: Exponential Mortality
149(1)
7.11 Value of Life Insurance: GoMa Mortality
149(1)
7.12 Life Insurance Paid by Installments
150(1)
7.13 NSP: Delayed and Term Insurance
150(1)
7.14 Variations on Life Insurance
151(3)
7.15 What If You Stop Paying Premiums?
154(3)
7.16 Duration of Life Insurance
157(2)
7.17 Following a Group of Policies
159(1)
7.18 The Next Generation: Universal Life Insurance
160(2)
7.19 Further Reading
162(1)
7.20 Notation
162(1)
7.21 Problems
162(2)
8 Models of DB vs. DC Pensions
164(21)
8.1 A Choice of Pension Plans
164(1)
8.2 The Core of Defined Contribution Pensions
165(4)
8.3 The Core of Defined Benefit Pensions
169(3)
8.4 What Is the Value of a DB Pension Promise?
172(4)
8.5 Pension Funding and Accounting
176(4)
8.6 Further Reading
180(1)
8.7 Notation
181(1)
8.8 Problems
182(3)
II WEALTH MANAGEMENT: APPLICATIONS AND IMPLICATIONS
9 Sustainable Spending at Retirement
185(30)
9.1 Living in Retirement
185(2)
9.2 Stochastic Present Value
187(3)
9.3 Analytic Formula: Sustainable Retirement Income
190(2)
9.4 The Main Result: Exponential Reciprocal Gamma
192(1)
9.5 Case Study and Numerical Examples
193(9)
9.6 Increased Sustainable Spending without More Risk?
202(4)
9.7 Conclusion
206(2)
9.8 Further Reading
208(1)
9.9 Problems
208(1)
9.10 Appendix: Derivation of the Formula
209(6)
10 Longevity Insurance Revisited
215(34)
10.1 To Annuitize or Not To Annuitize?
215(1)
10.2 Five 95-Year-Olds Playing Bridge
216(2)
10.3 The Algebra of Fixed and Variable Tontines
218(2)
10.4 Asset Allocation with Tontines
220(5)
10.5 A First Look at Self-Annuitization
225(1)
10.6 The Implied Longevity Yield
226(8)
10.7 Advanced-Life Delayed Annuities
234(7)
10.8 Who Incurs Mortality Risk and Investment Rate Risk?
241(3)
10.9 Further Reading
244(1)
10.10 Notation
245(1)
10.11 Problems
245(4)
III ADVANCED TOPICS
11 Options within Variable Annuities
249(21)
11.1 To Live and Die in VA
249(3)
11.2 The Value of Paying by Installments
252(5)
11.3 A Simple Guaranteed Minimum Accumulation Benefit
257(1)
11.4 The Guaranteed Minimum Death Benefit
258(1)
11.5 Special Case: Exponential Mortality
259(3)
11.6 The Guaranteed Minimum Withdrawal Benefit
262(6)
11.7 Further Reading
268(1)
11.8 Notation
269(1)
12 The Utility of Annuitization
270(23)
12.1 What Is the Protection Worth?
270(1)
12.2 Models of Utility, Value, and Price
271(1)
12.3 The Utility Function and Insurance
272(2)
12.4 Utility of Consumption and Lifetime Uncertainty
274(4)
12.5 Utility and Annuity Asset Allocation
278(3)
12.6 The Optimal Timing of Annuitization
281(1)
12.7 The Real Option to Defer Annuitization
282(5)
12.8 Advanced RODA Model
287(2)
12.9 Subjective vs. Objective Mortality
289(1)
12.10 Variable vs. Fixed Payout Annuities
290(1)
12.11 Further Reading
291(1)
12.12 Notation
292(1)
13 Final Words
293(2)
14 Appendix
295(6)
Bibliography 301(8)
Index 309

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